Actually, it is the solo project of former Deep Purple guitarist Ritchie Blackmore. It initially started as a quick off-shoot project of Blackmore (backed by Elf, a band fronted by Ronnie James Dio) because he wanted to record a cover of the Quatermass song "Black Sheep of the Family". After Blackmore left Deep Purple, the band became a full-time project. During the '70s, they played Hard Rock; in the early '80s they adopted more AOR-ish sound, briefly involving Graham Bonnett on vocals replacing Dio, before leaving Rainbow to form Alcatrazz. Bonnett would then be replaced by Joe Lynn Turner of Fandango fame.

Rainbow provides examples of:

"I see a rainbow rising" from "Stargazer" on Rising (also called Rainbow Rising because of that line and the way the title was written on the album cover).

Stranger in Us All gets name-dropped on the track "Black Masquerade".

Arena Rock: Mostly from Difficult to Cure onwards, though Down to Earth touched on it too.

Control Freak: Ritchie Blackmore, changing the line up whenever he felt like it. He especially seemed to dislike bassists. It also led to his legendarily adversarial relationship with Ronnie James Dio and ironically led to Dio becoming something of this himself due to his intense desire to never be controlled by anyone again.

Cover Version: "Black Sheep of the Family" (Quatermass), "Still I'm Sad" (The Yardbirds), and "Since You Been Gone" (Russ Ballard).

And if you were wondering, "I Surrender" by Russ Ballard was written by him for Rainbow, so it doesn't really count as a cover since they were the first to perform it.

Epic Rocking: "Stargazer", "A Light in the Black" and "Rainbow Eyes" are all over seven minutes long.

Their live shows during the Dio era count as well, they'd stretch most of the songs out to over 10 minutes.

Genre Shift: "If You Don't Like Rock 'n' Roll" on Ritchie Blackmore's Rainbow. For this song the album suddenly goes from Hard Rock / Proto-Metal to old school Rock and Roll with Little Richard esque piano riffs.

Heavy Mithril: Dio-era Rainbow is one of the earliest examples of the trope.

I Am the Band: Ritchie Blackmore's tendency towards this caused strife both in this band and in every other band he ever played with. Ironically enough, many people remember Rainbow for Ronnie James Dio's vocals and charismatic stage presence as much as Blackmore's guitar (Ronnie was even the only other member to stay in between the debut album and Rising).

Lead Bassist: Roger Glover. Ritchie Blackmore brought in his old Deep Purple bandmate to help write songs after Dio's departure and Ritchie's desire to conquer America (the three Dio-era albums sold poorly there despite tremendous critical acclaim). Glover would be the longest-tenured bandmember next to Blackmore himself, as well as writing and producing.

Lighter and Softer: Ritchie Blackmore was hell-bent on cracking the US charts, so the band adopted a pop rock / AOR oriented sound after the three Dio-fronted albums.

Mohs Scale of Rock and Metal Hardness: Their first album had a lot of 5's and 6's plus a few in the 3-4 range. Rising and Long Live Rock 'N' Roll are mostly in the 6-7 range with some stuff that goes down a few notches. Down to Earth is, again, mostly 5-6. The stuff from Difficult to Cure onwards is mostly in the 4-5 range with the odd jump up or down a notch.

New Sound Album: Difficult to Cure introduced the more commercial AOR sound for the '80s.

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